Related papers: Recent advances on the log-rank conjecture in comm…
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is…
Communication complexity is a fundamental aspect of information science, concerned with the amount of communication required to solve a problem distributed among multiple parties. The standard quantification of one-way communication…
We prove the existence of (one-way) communication tasks with a subconstant versus superconstant asymptotic gap, which we call "doubly infinite," between their quantum information and communication complexities. We do so by studying the…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the…
Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as…
This work studies distributed learning in the spirit of Yao's model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some…
In this paper, we consider distributed decision-making over stochastic communication links in multi-agent systems. We show how to extend the current literature on potential games with binary log-linear learning (which mainly focuses on…
In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality…
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995)…
A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and…
An open problem in communication complexity proposed by several authors is to prove that for every Boolean function f, the task of computing f(x AND y) has polynomially related classical and quantum bounded-error complexities. We solve a…
This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability…
Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive…
We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $\mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there…
Despite the extreme popularity of deep learning in science and industry, its formal understanding is limited. This thesis puts forth notions of rank as key for developing a theory of deep learning, focusing on the fundamental aspects of…
Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known "entanglement…
Two-party one-way quantum communication has been extensively studied in the recent literature. We target the size of minimal information that is necessary for a feasible party to finish a given combinatorial task, such as distinction of…